Question: The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=2+\frac{3n}{n+7}$. $\sum\limits_{n=1}^{8}{{{a}_{n}}}=$
$\sum\limits_{n=1}^{8}{{{a}_{n}}}~~~$ is the same as $~~~{{S}_{8}}\,$. ${{S}_{8}}=2+\frac{3\cdot 8}{15}=\frac{30+24}{15}=\frac{54}{15}=\frac{18}{5}$